Lehmer Pairs Revisited

ABSTRACT

We seek to understand how the technical definition of a Lehmer pair can be related to more analytic properties of the Riemann zeta function, particularly the location of the zeros of ζ′(s). Because we are interested in the connection [Csordas et al. 94] between Lehmer pairs and the de Bruijn–Newman constant Λ, we assume the Riemann hypothesis throughout. We define strong Lehmer pairs via an inequality on the derivative of the pre-Schwarzian of Riemann’s function Ξ(t), evaluated at consecutive zeros: Theorem 1 shows that strong Lehmer pairs are Lehmer pairs. Theorem 2 describes PΞ′(γ) in terms of ζ′(ρ) where ρ = 1/2 + iγ. Theorem 3 expresses PΞ′(γ₊) + PΞ′(γ₋) in terms of nearby zeros ρ′ of ζ′(s). We examine 114, 661 pairs of zeros of ζ(s) around height t = 10⁶, finding 855 strong Lehmer pairs. These are compared to the corresponding zeros of ζ′(s) in the same range.

KEYWORDS:

• Riemann hypothesis
• de Bruijn–Newman constant
• backward heat equation
• Lehmer pair
• random matrix theory

2000 AMS Subject Classification:

• 11M06
• 11M50
• 11Y35

Acknowledgments

We would like to thank David Farmer for sharing his computations of zeros of ζ′(s) in the range 10⁶ ≤ t ≤ 10⁶ + 6 · 10⁴ and the anonymous referee for a careful reading of the manuscript and helpful suggestions.

Notes

¹ Red–yellow–green–blue, as one travels around the origin counterclockwise.

² In [Dueñez et al. 10, Section 6], Y is assumed to be 0.

³ And with a slight bias to be negative, since the π/4 term gives Y a slight bias to be positive.

⁴ It would be desirable to bound the error in making this approximation.